In the grand canonical ensemble, the particle number N is not constant. It is, therefore, instructive to calculate the fluctuation in this quantity. As usual, this is defined to be
In order to calculate this derivative, it is useful to introduce the Helmholtz free energy per particle defined as follows:
where is the volume per particle.
The chemical potential is defined by
Similarly, the pressure is given by
But recall the definition of the isothermal compressibility:
and the relative fluctuation is given by
Therefore, in the thermodynamic limit, the particle number fluctuations vanish, and the grand canonical ensemble is equivalent to the canonical ensemble.